Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234345 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(17,18,23,29)(19,24,33,42)(20,34,36,26)(21,28,22,27)(25,38,44,35)(30,40,31,48)(32,45,37,41)(39,46,43,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,24,29,42)(20,31,36,30)(21,37,22,32)(25,43,44,39)(26,40,34,48)(27,45,28,41)(35,46,38,47);(9,11)(10,13)(12,16)(14,15)(17,20,23,36)(18,25,29,44)(19,30,33,31)(21,35,22,38)(24,39,42,43)(26,27,34,28)(32,46,37,47)(40,41,48,45);(3,8)(5,7)(11,16)(13,15)(17,20)(18,26)(19,31)(21,38)(22,35)(23,36)(24,40)(25,28)(27,44)(29,34)(30,33)(32,46)(37,47)(39,45)(41,43)(42,48);(2,6)(5,7)(10,14)(13,15)(17,21,23,22)(18,27,29,28)(19,32,33,37)(20,35,36,38)(24,41,42,45)(25,34,44,26)(30,46,31,47)(39,48,43,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7141552);